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Autoregressive
In statistics, econometrics and signal processing, an autoregressive (AR) model is a representation of a type of random process; as such, it is used to describe certain time-varying processes in nature, economics, etc. The autoregressive model specifies that the output variable depends linearly on its own previous values and on a stochastic term (an imperfectly predictable term); thus the model is in the form of a stochastic difference equation (or recurrence relation which should not be confused with differential equation). Together with the moving-average (MA) model, it is a special case and key component of the more general autoregressive–moving-average (ARMA) and autoregressive integrated moving average (ARIMA) models of time series, which have a more complicated stochastic structure; it is also a special case of the vector autoregressive model (VAR), which consists of a system of more than one interlocking stochastic difference equation in more than one evolving random vari ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Autoregressive–moving-average Model
In the statistical analysis of time series, autoregressive–moving-average (ARMA) models provide a parsimonious description of a (weakly) stationary stochastic process in terms of two polynomials, one for the autoregression (AR) and the second for the moving average (MA). The general ARMA model was described in the 1951 thesis of Peter Whittle, ''Hypothesis testing in time series analysis'', and it was popularized in the 1970 book by George E. P. Box and Gwilym Jenkins. Given a time series of data X_t, the ARMA model is a tool for understanding and, perhaps, predicting future values in this series. The AR part involves regressing the variable on its own lagged (i.e., past) values. The MA part involves modeling the error term as a linear combination of error terms occurring contemporaneously and at various times in the past. The model is usually referred to as the ARMA(''p'',''q'') model where ''p'' is the order of the AR part and ''q'' is the order of the MA part (as defined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moving-average Model
In time series analysis, the moving-average model (MA model), also known as moving-average process, is a common approach for modeling univariate time series. The moving-average model specifies that the output variable is cross-correlated with a non-identical to itself random-variable. Together with the autoregressive (AR) model, the moving-average model is a special case and key component of the more general ARMA and ARIMA models of time series, which have a more complicated stochastic structure. The moving-average model should not be confused with the moving average, a distinct concept despite some similarities. Contrary to the AR model, the finite MA model is always stationary. Definition The notation MA(''q'') refers to the moving average model of order ''q'': : X_t = \mu + \varepsilon_t + \theta_1 \varepsilon_ + \cdots + \theta_q \varepsilon_ = \mu + \sum_^q \theta_i \varepsilon_ + \varepsilon_, where \mu is the mean of the series, the \theta_1,...,\theta_q are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Autoregressive Integrated Moving Average
In statistics and econometrics, and in particular in time series analysis, an autoregressive integrated moving average (ARIMA) model is a generalization of an autoregressive moving average (ARMA) model. Both of these models are fitted to time series data either to better understand the data or to predict future points in the series (forecasting). ARIMA models are applied in some cases where data show evidence of non-stationarity in the sense of mean (but not variance/ autocovariance), where an initial differencing step (corresponding to the "integrated" part of the model) can be applied one or more times to eliminate the non-stationarity of the mean function (i.e., the trend). When the seasonality shows in a time series, the seasonal-differencing could be applied to eliminate the seasonal component. Since the ARMA model, according to the Wold's decomposition theorem, is theoretically sufficient to describe a regular (a.k.a. purely nondeterministic) wide-sense stationary time se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gilbert Walker (physicist)
Sir Gilbert Thomas Walker (14 June 1868 – 4 November 1958) was an English physicist and statistician of the 20th century. Walker studied mathematics and applied it to a variety of fields including aerodynamics, electromagnetism and the analysis of time-series data before taking up a teaching position at the University of Cambridge. Although he had no experience in meteorology, he was recruited for a post in the Indian Meteorological Department where he worked on statistical approaches to predict the monsoons. He developed the methods in the analysis of time-series data that are now called the Yule-Walker equations. He is known for his groundbreaking description of the Southern Oscillation, a major phenomenon of global climate, and for discovering what is named after him as the Walker circulation, and for greatly advancing the study of climate in general. He was also instrumental in aiding the early career of the Indian mathematical prodigy, Srinivasa Ramanujan. Earl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Backshift Operator
In time series analysis, the lag operator (L) or backshift operator (B) operates on an element of a time series to produce the previous element. For example, given some time series :X= \ then : L X_t = X_ for all t > 1 or similarly in terms of the backshift operator ''B'': B X_t = X_ for all t > 1. Equivalently, this definition can be represented as : X_t = L X_ for all t \geq 1 The lag operator (as well as backshift operator) can be raised to arbitrary integer powers so that : L^ X_ = X_ and : L^k X_ = X_. Lag polynomials Polynomials of the lag operator can be used, and this is a common notation for ARMA (autoregressive moving average) models. For example, : \varepsilon_t = X_t - \sum_^p \varphi_i X_ = \left(1 - \sum_^p \varphi_i L^i\right) X_t specifies an AR(''p'') model. A polynomial of lag operators is called a lag polynomial so that, for example, the ARMA model can be concisely specified as : \varphi (L) X_t = \theta (L) \varepsilon_t where \varphi (L) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Udny Yule
George Udny Yule FRS (18 February 1871 – 26 June 1951), usually known as Udny Yule, was a British statistician, particularly known for the Yule distribution. Personal life Yule was born at Beech Hill, a house in Morham near Haddington, Scotland and died in Cambridge, England. He came from an established Scottish family composed of army officers, civil servants, scholars, and administrators. His father, Sir George Udny Yule (1813–1886) was a brother of the noted orientalist Sir Henry Yule (1820–1889). His great uncle was the botanist John Yule. In 1899, Yule married May Winifred Cummings. The marriage was annulled in 1912, producing no children.annulment: Yates, 1952 Education and teaching Udny Yule was educated at Winchester College and at the age of 16 at University College London where he read engineering. After a year in Bonn doing research in experimental physics under Heinrich Rudolf Hertz, Yule returned to University College in 1893 to work as a demonst ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Autocovariance
In probability theory and statistics, given a stochastic process, the autocovariance is a function that gives the covariance of the process with itself at pairs of time points. Autocovariance is closely related to the autocorrelation of the process in question. Auto-covariance of stochastic processes Definition With the usual notation \operatorname for the expectation operator, if the stochastic process \left\ has the mean function \mu_t = \operatorname _t/math>, then the autocovariance is given by where t_1 and t_2 are two moments in time. Definition for weakly stationary process If \left\ is a weakly stationary (WSS) process, then the following are true: :\mu_ = \mu_ \triangleq \mu for all t_1,t_2 and :\operatorname [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinite Impulse Response
Infinite impulse response (IIR) is a property applying to many linear time-invariant systems that are distinguished by having an impulse response h(t) which does not become exactly zero past a certain point, but continues indefinitely. This is in contrast to a finite impulse response (FIR) system in which the impulse response ''does'' become exactly zero at times t>T for some finite T, thus being of finite duration. Common examples of linear time-invariant systems are most electronic and digital filters. Systems with this property are known as ''IIR systems'' or ''IIR filters''. In practice, the impulse response, even of IIR systems, usually approaches zero and can be neglected past a certain point. However the physical systems which give rise to IIR or FIR responses are dissimilar, and therein lies the importance of the distinction. For instance, analog electronic filters composed of resistors, capacitors, and/or inductors (and perhaps linear amplifiers) are generally IIR filter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy Distribution
The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) function, or Breit–Wigner distribution. The Cauchy distribution f(x; x_0,\gamma) is the distribution of the -intercept of a ray issuing from (x_0,\gamma) with a uniformly distributed angle. It is also the distribution of the ratio of two independent normally distributed random variables with mean zero. The Cauchy distribution is often used in statistics as the canonical example of a "pathological" distribution since both its expected value and its variance are undefined (but see below). The Cauchy distribution does not have finite moments of order greater than or equal to one; only fractional absolute moments exist., Chapter 16. The Cauchy distribution has no moment generating function. In mathematics, it is closely related to the P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Philosophical Transactions Of The Royal Society
''Philosophical Transactions of the Royal Society'' is a scientific journal published by the Royal Society. In its earliest days, it was a private venture of the Royal Society's secretary. It was established in 1665, making it the first journal in the world exclusively devoted to science, and therefore also the world's longest-running scientific journal. It became an official society publication in 1752. The use of the word ''philosophical'' in the title refers to natural philosophy, which was the equivalent of what would now be generally called ''science''. Current publication In 1887 the journal expanded and divided into two separate publications, one serving the physical sciences ('' Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences'') and the other focusing on the life sciences ('' Philosophical Transactions of the Royal Society B: Biological Sciences''). Both journals now publish themed issues and issues resulting from pap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
White Noise
In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. The term is used, with this or similar meanings, in many scientific and technical disciplines, including physics, acoustical engineering, telecommunications, and statistical forecasting. White noise refers to a statistical model for signals and signal sources, rather than to any specific signal. White noise draws its name from white light, although light that appears white generally does not have a flat power spectral density over the visible band. In discrete time, white noise is a discrete signal whose samples are regarded as a sequence of serially uncorrelated random variables with zero mean and finite variance; a single realization of white noise is a random shock. Depending on the context, one may also require that the samples be independent and have identical probability distribution (in other words independent and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Notation
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate is . An example with three indeterminates is . Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry. Etymology The word ''polynomial'' j ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
